A 100 amino acid folded protein (F) is in equilibrium with it’s unfolded (U) (or
denatured) form. Let ΔrH(std) =+347 kJ mol-1 and ΔrS(std) =-1010 J K-1 mol-1 for the equilibrium () at 25℃.
(a) Calculate the concentration of protein in each form if the total
concentration of protein is 10μM.
(b)Assuming that ΔrH(std) is independent of temperature, calculate the equilibrium constant at 37℃ using the van’t Hoff equation.
To calculate the concentration of protein in each form and the equilibrium constant at different temperatures, we can use the principles of thermodynamics and the van't Hoff equation. Here's how you can do it:
(a) Calculate the concentration of protein in each form:
1. First, convert the given temperature from Celsius to Kelvin:
T = 25 + 273.15 = 298.15 K
2. Calculate ΔG(std) (standard Gibbs free energy change) using the equation:
ΔG(std) = ΔrH(std) - T * ΔrS(std)
ΔG(std) = 347 kJ/mol - 298.15 K * (-1010 J K-1 mol-1)
= 347 kJ/mol + 301636.5 J/mol
= 647 kJ/mol
3. Calculate the equilibrium constant (K) using the equation:
K = e^(-ΔG(std) / (R * T))
R is the gas constant (8.314 J/(mol⋅K))
K = e^(-647000 J/mol / (8.314 J/(mol⋅K) * 298.15 K))
≈ 2.867 x 10^-4
4. Calculate the concentration of the unfolded form (U) using the equation:
[U] = (K / (1 + K)) * [Total Protein]
[Total Protein] = 10 μM (given)
[U] = (2.867 x 10^-4 / (1 + 2.867 x 10^-4)) * 10 μM
≈ 0.0002867 μM
5. Calculate the concentration of the folded form (F) using the equation:
[F] = [Total Protein] - [U]
[F] = 10 μM - 0.0002867 μM
≈ 9.9997 μM
Therefore, the concentration of the protein in the unfolded form (U) is approximately 0.0002867 μM, and the concentration in the folded form (F) is approximately 9.9997 μM.
(b) Calculate the equilibrium constant at 37℃ using the van't Hoff equation:
1. Convert the temperature from Celsius to Kelvin:
T = 37 + 273.15 = 310.15 K
2. Use the van't Hoff equation:
ln(K₂/K₁) = (ΔH(std) / R) * ((1/T₁) - (1/T₂))
ΔH(std) = ΔrH(std) (given) = 347 kJ/mol
R = 8.314 J/(mol⋅K)
T₁ = 298.15 K (given)
T₂ = 310.15 K (calculated)
3. Solve for ln(K₂/K₁):
ln(K₂/K₁) = (347 kJ/mol / (8.314 J/(mol⋅K)) * ((1/298.15 K) - (1/310.15 K))
≈ 0.161
4. Solve for K₂/K₁ by taking the exponential of both sides:
K₂/K₁ = e^(ln(K₂/K₁))
= e^(0.161)
≈ 1.175
Therefore, the equilibrium constant (K₂) at 37℃ is approximately 1.175 times the equilibrium constant (K₁) at 25℃.